Small and large-amplitude gravitational instability of an elastically compressible viscoelastic Maxwell solid overlying an inviscid incompressible fluid: Dependence of growth rates on wave number and elastic constants at low Deborah numbers

To understand how elasticity affects convective instability of viscoelastic fluids near the viscous limit, we carried out numerical experiments of Rayleigh–Taylor instability of compressible viscoelastic fluids overlying inviscid inelastic substrata (hence with infinite viscosity and elasticity ratios). Unlike incompressible viscous fluids, for which growth becomes super-exponential when perturbations to the thickness of the unstable layer grow to several tens of percent of the thickness, for compressible viscoelastic fluids, super-exponential growth does not appear to develop at relatively low Deborah numbers, De = Δρgh/η ∼ 10− 6–10− 4 (where Δρ is the density difference between unstable layer and substratum, g is gravity, h is the thickness of the unstable layer, and η is its viscosity) and for very high (> 106) ratios of viscosity between layer and substratum, which characterize large-scale geodynamic systems. This behavior differs from that of viscoelastic two-layer systems with higher Deborah numbers (> 10− 4) and with smaller viscosity ratios (< 104) because, instead of accelerating the instability, as for incompressible media, elastic deformation may also retard growth in its most rapid phase as the amplitude of flow increases. For small De (< 10− 3–10− 4), retardation of growth manifests itself in three ways: (1) while perturbations remain small, the commonly observed exponential growth is delayed, (2) during exponential growth, the growth rate decreases monotonically with decreasing De, and (3) when perturbations grow to large amplitude (> 100%), the exponential growth rate decreases, due to the formation of a compressible viscoelastic drop that has a distinguishable drop head and a stretched filament. Values of De appropriate for Earth's mantle suggest that in most circumstances elasticity will not affect the growth of lithospheric instabilities, but for high density contrasts (e.g. atypically warm lithosphere) elasticity may retard their growth. By contrast, for relatively large Deborah numbers (> 10− 3), finite viscosity ratios (< 107), and small amplitudes of perturbations, elasticity accelerates growth of the instability.